A sample of 64 statistics students at a small college had a mean mathematics ACT score of 28 with standard deviation of 4. Estimate the mean mathematics ACT score for all statistics students at college. Give the 95% confidence interval.
95% = mean ± 1.96 SEm
SEm = SD/√n
To estimate the mean mathematics ACT score for all statistics students at the college and calculate the 95% confidence interval, we can use the formula for confidence interval for the population mean:
Confidence Interval = Sample Mean ± Margin of Error
1. Calculate the Margin of Error:
The margin of error is determined by the confidence level and the standard deviation of the population. For a 95% confidence level, we will use a Z-score of 1.96 (which corresponds to the 95% confidence interval).
Margin of Error = Z * (Standard Deviation / √Sample Size)
In this case:
Z = 1.96
Standard Deviation = 4 (given in the question)
Sample Size = 64 (provided in the question)
Margin of Error = 1.96 * (4 / √64)
2. Calculate the Confidence Interval:
The confidence interval is calculated by taking the sample mean and adding/subtracting the margin of error.
Confidence Interval = Sample Mean ± Margin of Error
In this case:
Sample Mean = 28 (given in the question)
Confidence Interval = 28 ± Margin of Error
Now, substituting the values we have:
Confidence Interval = 28 ± (1.96 * (4 / √64))
Simplifying further:
Confidence Interval = 28 ± (1.96 * (4 / 8))
Confidence Interval = 28 ± (1.96 * 0.5)
Confidence Interval = 28 ± 0.98
Therefore, the 95% confidence interval for the mean mathematics ACT score for all statistics students at the college is from 27.02 to 28.98.